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Beliebt Rechnen >

integral of x(x+1)^9

Lösung

\int x (x + 1)^{9} d x

Lösung

\frac{1}{110} (x + 1)^{10} (10 x - 1) + C

Schritte zur Lösung

\int x (x + 1)^{9} d x

Wende U-Substitution an

= \int \frac{9 u ( 9 u - 1 )}{9} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{9} \cdot \int 9 u (9 u - 1) d u

Multipliziere aus

9 u (9 u - 1) : u^{\frac{2}{9}} - 9 u

= \frac{1}{9} \cdot \int u^{\frac{2}{9}} - 9 u d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{9} (\int u^{\frac{2}{9}} d u - \int 9 u d u)

\int u^{\frac{2}{9}} d u = \frac{9}{11} u^{\frac{11}{9}}

\int 9 u d u = \frac{9}{10} u^{\frac{10}{9}}

= \frac{1}{9} (\frac{9}{11} u^{\frac{11}{9}} - \frac{9}{10} u^{\frac{10}{9}})

Setze in

u = (x + 1)^{9}

ein

= \frac{1}{9} (\frac{9}{11} ((x + 1)^{9})^{\frac{11}{9}} - \frac{9}{10} ((x + 1)^{9})^{\frac{10}{9}})

Vereinfache

\frac{1}{9} (\frac{9}{11} ((x + 1)^{9})^{\frac{11}{9}} - \frac{9}{10} ((x + 1)^{9})^{\frac{10}{9}}) : \frac{1}{110} (x + 1)^{10} (10 x - 1)

= \frac{1}{110} (x + 1)^{10} (10 x - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{110} (x + 1)^{10} (10 x - 1) + C

Graph

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